OFFSET
0,3
COMMENTS
The number of triples of standard tableaux of the same shape of height less than or equal to 2. - Mike Zabrocki, Mar 29 2007
From Peter Bala, Mar 20 2023: (Start)
For r a positive integer define S(r,n) = Sum_{k = 0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r. The present sequence is {S(3,n)}. For other cases see A361887 ({S(5,n)}) and A361890 ({S(7,n)}).
Gould (1974) proposed the problem of showing that S(3,n) was always divisible by S(1,n). See A183069 for {S(3,n)/S(1,n)}. In fact, calculation suggests that if r is odd then S(r,n) is always divisible by S(1,n).
Conjecture: Let b(n) = a(2*n-1). Then the supercongruence b(n*p^k) == b(n*p^(k-1)) (mod p^(3*k)) holds for positive integers n and k and all primes p >= 5. (End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1116
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
H. W. Gould, Problem E2384, Amer. Math. Monthly, 81 (1974), 170-171.
FORMULA
a(n) = Sum_{k=0..n} A120730(n,k)^3. - Philippe Deléham, Oct 18 2008
G.f.: hypergeometric expression with an anti-derivative, see Maple program. - Mark van Hoeij, May 06 2013
Recurrence: n*(n+1)^3*(7*n^2 - 14*n + 3)*a(n) = - n*(7*n^5 - 112*n^4 + 206*n^3 + 8*n^2 - 125*n + 48)*a(n-1) + 16*(n-1)*(28*n^5 - 133*n^4 + 194*n^3 - 33*n^2 - 120*n + 61)*a(n-2) + 64*(n-2)^3*(n-1)*(7*n^2 - 4)*a(n-3). - Vaclav Kotesovec, Mar 06 2014
a(n) ~ 2^(3*n+9/2) / (9 * Pi^(3/2) * n^(5/2)). - Vaclav Kotesovec, Mar 06 2014
a(n) = Sum_{j=0..floor(n/2)} A008315(n,j)^3. - Alois P. Heinz, Oct 17 2022
MAPLE
ogf := ((8*x-1)*(8*x+1)*hypergeom([1/4, 1/4], [1], 64*x^2)^2/(x+1)-3*Int((16*x-5)*hypergeom([1/4, 1/4], [1], 64*x^2)^2/(x+1)^2, x)+1)/(16*x);
series(ogf, x=0, 30); # Mark van Hoeij, May 06 2013
MATHEMATICA
Table[Sum[(Binomial[n, k]-Binomial[n, k-1])^3, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 06 2014 *)
PROG
(PARI) a(n)=sum(k=0, n\2, (binomial(n, k)-binomial(n, k-1))^3) /* Michael Somos, Jun 02 2005 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved